What is the net for a parallelepiped?

by Colin Foster

Visuospatial problems can be a great leveller in younger pupils I have pushed a stack of paper sideways mathematics. There are people with fancy mathematics to illustrate shearing of a rectangle into a parallelogram with preservation of area, but the focus here is on the three-dimensional solids sketched from different angles parallelogram face rather than the parallelepiped as a aredegrees the same who or find different, it very and difficult yet there to tellare whetherschool pupils two whole. So how do you make a net for a parallelepiped? who can do this sort of thing just by staring – and think If you’ve never tried to draw a net for a parallelepiped, that it’s obvious! So when these sorts of problems arise you might like to explore that now, before (or instead of) in the classroom the gap between what the pupil can do reading the rest of this article! and what the teacher can do can be quite small, if it exists at all! One of my favourite bits of project work with Year 8 pupils is to build a shape sorter – those toys that toddlers play the holes (Note 1). Designing a shape sorter that does that canwith, be where quite eachchallenging, solid fits as through it’s very one important and only that one noof Fig. 1 shape will go through the wrong hole – we don’t want to What usually happens A is parallelepipedthat pupils begin by drawing confuse those poor little toddlers! Typically, pupils go for the front parallelogram (shaded light grey in Figure things like a cube, a non-cube cuboid, a cylinder, a right- 1) and then pause for a while, wondering whether angled triangular prism and an equilateral triangular the parallelogram on the top surface can/must be prism. An extra constraint can be that all of the solids angles need to be different. Pupils attempting this will All sorts of interesting questions can arise, such as “Can alreadycongruent know to that the a first cuboid parallelogram, net can be made or whether as shown its youmust have fit insidemore than the container one cylinder?” (e.g. aor shoe “Can box) you atmake once. a in Figure 2, so they tend to try to adapt this by turning shape sorter just using cuboids?” I have found that more adventurous pupils tend to go for many-sided prisms or pyramids, such as hexagonal-based prisms, which tend to isbe to fiddly make to a non-cuboidconstruct but parallelepiped. not conceptually much more difficult. So perhaps a more interesting direction to go in Pupils always think that ‘parallelepiped’ is a very funny name, and I often wonder if I am saying it correctly! A parallelepiped (Fig. 1) is a prism whose faces are all parallelograms. It includes the special case of a cuboid (and thus a cube), but in general the parallelograms do not need to be rectangles. These shapes don’t come up very much in school until the sixth form, when some students will meet them when working with 3 by 3 matrices and the scalar triple product. Sometimes with Fig. 2

Net of a square cuboid

38 Mathematics in School, November 2017 The MA website www.m-a.org.uk A B

C D

Fig. 3

Do any of these fold up to make a parallelepiped? the rectangles into parallelograms, which is a sensible way to start. (Isometric paper can be useful.) Figure 3 shows some attempts – do any of them fold up to make a parallelepiped? Why or why not? (Note 2) Fig. 4 One way to deal with this (if you’re totally desperate) is to cut out the separate parallelograms that you want to A general parallelogram and its mirror image Notes with sticky tape to make the parallelepiped, and then cut ithave open on along your some finished edges parallelepiped, and look at what fit them you have together got! 1.2. IThe first way heard in ofwhich the idea some from of Keiththese Proffitt.images jump out of the page as But that is very much a last resort if you can’t get there three-dimensional can interfere with the visualization necessary to any other way! imagine them folded up. The crucial observation that helps is that opposite faces, 3. In general, the faces of a parallelogram are chiral (non-superimposable mirror images), even though the entire parallelepiped is not. when viewed from the outside, must be mirror images (Note 3). This may be counterintuitive, because when we 4. A lovely website for producing nets of a parallelepiped is available at: www.templatemaker.nl/index.php?template=parallelepiped&lang look at the parallelepiped from one direction, as in Figure 1, we see both shaded parallelograms on top of each other, and the same way round. But when you unfold the Reference

Foster, C. 2017 ‘A Fitting Challenge’, Teach Secondary, 6, 6, pp. 48–49. the reverse side of one of these, and that is why they have Available free at https://www.teachwire.net/news/ks3-maths- tonet be and mirror lay it imagesflat, one when of them drawn has onto turnthe net. over, In so a youcuboid, see lesson-plan-promote-understanding-of-3d-shapes-by-making- a-childr parallelogram.this reflecting makesSo, in Figure no difference, 4, a rotation because of 180° a rectangle will not transformis its own one reflection, of these but parallelograms this is not true into for the a generalother – and this may be surprising to pupils. This is an important insight, but there are other aspects to consider as well Keywords:  Visuospatial problems. Net; Parallelepiped; Parallelogram; to get a working parallelepiped net. Can you find all the Author Colin Foster, School of Education, University of for a net for a parallelepiped? (Note 4) Nottingham, Jubilee Campus, Wollaton Road, Nottingham possible nets, or state necessary and sufficient criteria NG8 1BB. For a free lesson plan relating to this lesson, see Foster e-mail: [email protected] (2017). website: www.foster77.co.uk

Mathematics in School, November 2017 The MA website www.m-a.org.uk 39